The ternary Goldbach problem asks whether every odd integer greater than 5 is the sum of three primes, and has its origins in a letter from Goldbach to Euler in 1742. Around 1910, Hardy and Littlewood developed the Circle Method- equivalent to Fourier Analysis over Z- to attack these types of additive problems, and achieved results under the Generalized Riemann Hypothesis. Finally, in 1937 Vinogradov used this method with some new ideas to unconditionally prove Ternary Goldbach for suffiently large odd integers.

We will give an exposition of Vinogradov's result and the tools involved, particularly the Circle Method, which are also the basic tools on which Helfgott's recent proof, of Ternary Goldbach for all odd integers, begins.